Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 13 \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 4·7-s + 8-s + 9-s − 10-s + 12-s + 13-s − 4·14-s − 15-s + 16-s + 17-s + 18-s − 4·19-s − 20-s − 4·21-s + 24-s + 25-s + 26-s + 27-s − 4·28-s + 6·29-s − 30-s − 4·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6630\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6630} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6630,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.44779945161218, −16.69954942719303, −16.18979435848618, −15.79729877368303, −15.15147396553094, −14.66080531937809, −13.95081786605414, −13.32488272556021, −12.89168283255828, −12.35540554857229, −11.80067827307953, −10.84316895418155, −10.36194548199416, −9.648570313350306, −8.965241881366715, −8.334282834281226, −7.476144836262978, −6.897534706402619, −6.227112294110476, −5.654968102077304, −4.427337478542937, −4.049257864368321, −3.041017836225429, −2.817038403094119, −1.481239375991468, 0, 1.481239375991468, 2.817038403094119, 3.041017836225429, 4.049257864368321, 4.427337478542937, 5.654968102077304, 6.227112294110476, 6.897534706402619, 7.476144836262978, 8.334282834281226, 8.965241881366715, 9.648570313350306, 10.36194548199416, 10.84316895418155, 11.80067827307953, 12.35540554857229, 12.89168283255828, 13.32488272556021, 13.95081786605414, 14.66080531937809, 15.15147396553094, 15.79729877368303, 16.18979435848618, 16.69954942719303, 17.44779945161218

Graph of the $Z$-function along the critical line